This paper contains an account, in abstract terms, of sufficiency and of its role in statistical decision problems. The study of sufficiency in abstract terms was initiated by Halmos and Savage [1], and the present paper, although self-contained, is to be regarded as a continuation of their work. The main objects of the paper are to show that the justification for the use of sufficient statistics in statistical methodology which is sketched in the final section of [1] is valid under certain quite general conditions, and to extend this justification to the case of sequential experiments. The paper falls into two parts of which the first (Sections 2-7) is mainly expository and provides an account of the theory of sufficiency in the nonsequential case. The second part (Sections 8-11) then extends the theory to sequential experiments.

Asymptotic properties are established for the Robbins-Monro [1] procedure of stochastically solving the equation $M(x) = \alpha$. Two disjoint cases are treated in detail. The first may be called the "bounded" case, in which the assumptions we make are similar to those in the second case of Robbins and Monro. The second may be called the "quasi-linear" case which restricts $M(x)$ to lie between two straight lines with finite and nonvanishing slopes but postulates only the boundedness of the moments of $Y(x) - M(x)$ (see Sec. 2 for notations). In both cases it is shown how to choose the sequence $\{a_n\}$ in order to establish the correct order of magnitude of the moments of $x_n - \theta$. Asymptotic normality of $a^{1/2}_n(x_n - \theta)$ is proved in both cases under a further assumption. The case of a linear $M(x)$ is discussed to point up other possibilities. The statistical significance of our results is sketched.

Theorems already enunciated in a previous paper on quadratic forms are used to determine the effects of inequality of variance and first order serial correlation of errors in the two-way classification on the analysis of variance. It is found that when the appropriate null hypothesis is true, inequality of variance from column to column results in an increased chance of exceeding the significance point for the test on homogeneity of column means, and a decreased chance for the corresponding test on row means. For moderate differences in variance neither effect is large. First order serial correlation within rows produces a large effect on the "between rows" comparisons, but little effect on the "between columns" comparisons.

Brown and Mood [1], [4] have recently given convenient distribution-free methods of testing and setting up confidence regions for the parameters of a linear regression model. Their technique, which is based on the use of medians, allows the parameters to be considered singly or simultaneously as required. Theil [5] gives two methods of constructing confidence intervals for single parameters, a "complete" method using rank correlation which is valid under the conditions assumed by Brown and Mood, and an "incomplete" method valid under wider conditions but not making full use of the data. For several parameters simultaneously, he obtains confidence regions in the weak sense of covering the true parameter point with probability not less than an assigned value. In the present paper we give a new distribution-free test for the hypothesis that all regression parameters have specified values, assuming only that the residuals are independent and have probability $\frac{1}{2}$ of being positive or negative. It can be used to set up exact confidence regions for the true parameter point. The new test avoids a defect which is shown to appear in the corresponding Brown and Mood test when the sample is not large. The distribution of the test statistic is found explicitly only for the case of two parameters, though in principle the idea extends to any number of parameters. The presence of repeated values of the independent variable necessitates certain modifications in the test, and a method of computing the appropriate distributions in such cases is described.

In this paper the following asymptotic efficiencies are computed for the given two-sample tests against normal alternatives to the null hypothesis: $\text{rank test for location}\ldots\ldots\ldots\ldots 3/\pi \cong 95{\tt\%}$ $\text{median test for location}\ldots\ldots\ldots\ldots 2/\pi \cong 64{\tt\%}$ $\text{run test for location}\ldots\ldots\ldots \ldots 0$ $\text{run test for dispersion}\ldots\ldots\ldots\ldots 0$ $\text{square rank test for dispersion}\ldots\ldots\ldots\ldots 15/2\pi^2 \cong 76{\tt\%}$ Also, general expressions for means and variances of some of these test criteria are found for distributions alternative to the null hypothesis.

By a generalisation of a theorem by Doob, certain confidence or acceptance contours for distribution functions are obtained. The power of tests based on such contours is briefly discussed and some approximate results derived. Using the aforementioned generalisation of Doob's theorem, the limiting joint probability distribution of the coordinates of the maximum deviation between a sample distribution and the corresponding parent distribution is evaluated.

A method is given for constructing confidence bands for polynomial growth-type curves. The method assumes that the mean population size can be expressed as a polynomial in time and that the generalized $T$ function for the mean values of the observations at fixed time points possesses a known and parameter-free distribution. Independence between observations at various times is not assumed. The method yields only a lower bound for the confidence coefficient.

A hypothesis often made about a sequence of real-valued r.v. (random variables), $\{X_n, n = 0, \pm 1, \pm 2, \cdots\}$ is that there exist certain real constants $\alpha_1, \alpha_2, \cdots, \alpha_k$ such that if we write \begin{equation*}\tag{1} Y_n = X_n + \alpha_1X_{n-1} + \cdots + \alpha_kX_{n-k},\end{equation*} then $\{Y_n\}$ is a sequence of independent r.v. Now, very often in practice, the observed sequence $\{X_n\}$ consists in observations made at equidistant $t$-points on a stochastic process with a continuous parameter $t$. Restricting our attention to the case $k = 1$, we then have a r.f. (random function) $X(t)$, defined for all $t$ in an interval, with the following property: There exist a value of $t (t_0$, say), and $h > 0$, and a real number $\alpha$ such that a hypothesis of the type mentioned is satisfied by the sequence $\{Y(t_0, h; n), n = 0, \pm1, \pm2, \cdots\}$, where \begin{equation*}\tag{2} Y(t_0, h; n) = X(t_0 + nh) - \alpha X(t_0 + \lbrack n - 1\rbrack h).\end{equation*} But if such a hypothesis is true for one value of $h$, it is not necessarily so for some other value; and we have to make the additional assumption that the particular length of the $t$-intervals with which we are concerned is precisely the one for which the hypothesis holds. This assumption may not be reasonable in every case, Instead we may wish to work with a hypothesis similar to the above, but which holds for all positive $h$ in some interval. In this paper, we investigate the existence and form of random functions satisfying a hypothesis of this type. Section 1 contains a statement of the problem and some simple results. It turns out that any random function possessing the required property can be expressed as the product of an exponential function of $t$ and a r.f. with independent increments. Section 2 deals with the limit in distribution of a sequence of Stieltjes approximating sums involving a r.f. with independent increments. Finally, in Section 3, these results are applied to the problem under investigation, and further possibilities are investigated. It must be noted that, in this investigation, we are concerned only with results in distribution. That is to say, we are dealing throughout only with parametric families of probability laws and talking in terms of r.f. merely for convenience.

It is frequently desirable on practical grounds to terminate a life test by a preassigned time $T_0$. In this paper we consider life tests which are truncated as follows. With $n$ items placed on test, it is decided in advance that the experiment will be terminated at $\min (X_{r0,n}, T_0)$, where $X_{r0,n}$ is a random variable equal to the time at which the $r_0$th failure occurs and $T_0$ is a truncation time, beyond which the experiment will not be run. Both $r_0$ and $T_0$ are assigned before experimentation starts. If the experiment is terminated at $X_{r0,n}$ (that is, if $r_0$ failures occur before time $T_0$), then the action in terms of hypothesis testing is the rejection of some specified null-hypothesis. If the experiment is terminated at time $T_0$ (that is, if the $r_0$th failure does not occur before time $T_0$), then the action in terms of hypothesis testing is the acceptance of some specified null-hypothesis. While truncated procedures can be considered for any life distribution, we limit ourselves here to the case where the underlying life distribution is specified by a p.d.f. of the exponential form, $f(x; \theta) = \theta^{-1}e^{-x/\theta}, x > 0, \theta > 0$. The practical justification for using this kind of distribution as a first approximation to a number of test situations is discussed in a recent paper by Davis [1]. It is a common assumption for electron tube life. Two situations are considered. The first is the nonreplacement case in which a failure occurring during the test is not replaced by a new item. The second is the replacement case where failed items are replaced at once by new items drawn at random from the same p.d.f. as the original $n$ items. Formulae are given for $E_\theta(r)$, the expected number of observations to reach a decision; for $E_\theta(T)$, the expected waiting time to reach a decision; and for $L(\theta)$, the probability of accepting the hypothesis that $\theta = \theta_0$, the value associated with the null-hypothesis, when $\theta$ is the true value. Some procedures are worked out for finding truncated tests meeting specified conditions, and practical illustrations are given. It is an intrinsic feature of all life test decision procedures that they are in some sense truncated, although not necessarily by a fixed time $T_0$. In Section 3 we give exact formulae for $E_\theta(r)$ and $E_\theta(T)$ for a decision procedure given in [2]. There is a close relation between these results and those in Section 2.

This paper deals with the distribution of any observation, $x_i$, in an ordered sample of size $n$ from a normal population with zero mean and unit standard deviation. The distribution has been developed as a series of Gamma functions, and has been used to obtain the distribution of $q_i = (x_i/s)$, where $s$ is an independent estimate of the standard deviation with $\nu$ degrees of freedom. In a similar manner the distribution of the Studentized maximum modulus $u_n = | x_n/s |$ has been obtained and upper 5 per cent points of $q_n$ and upper and lower 5 per cent points of $u_n$ have been given. The method of obtaining the different distributions essentially depends on appropriate expansions of the normal probability integral and its powers in the intervals $- \infty$ to $x$ and 0 to $x$.

A classical result due to Wilks [1] on the distribution of the likelihood ratio $\lambda$ is the following. Under suitable regularity conditions, if the hypothesis that a parameter $\theta$ lies on an $r$-dimensional hyperplane of $k$-dimensional space is true, the distribution of $-2 \log \lambda$ is asymptotically that of $\chi^2$ with $k - r$ degrees of freedom. In many important problems it is desired to test hypotheses which are not quite of the above type. For example, one may wish to test whether $\theta$ is on one side of a hyperplane, or to test whether $\theta$ is in the positive quadrant of a two-dimensional space. The asymptotic distribution of $-2 \log \lambda$ is examined when the value of the parameter is a boundary point of both the set of $\theta$ corresponding to the hypothesis and the set of $\theta$ corresponding to the alternative. First the case of a single observation from a multivariate normal distribution, with mean $\theta$ and known covariance matrix, is treated. The general case is then shown to reduce to this special case where the covariance matrix is replaced by the inverse of the information matrix. In particular, if one tests whether $\theta$ is on one side or the other of a smooth $(k - 1)$-dimensional surface in $k$-dimensional space and $\theta$ lies on the surface, the asymptotic distribution of $\lambda$ is that of a chance variable which is zero half the time and which behaves like $\chi^2$ with one degree of freedom the other half of the time.

The usual test that a sample comes from a distribution of given form is performed by counting the number of observations falling into specified cells and applying the $\chi^2$ test to these frequencies. In estimating the parameters for this test, one may use the maximum likelihood (or equivalent) estimate based (1) on the cell frequencies, or (2) on the original observations. This paper shows that in (2), unlike the well known result for (1), the test statistic does not have a limiting $\chi^2$-distribution, but that it is stochastically larger than would be expected under the $\chi^2$ theory. The limiting distribution is obtained and some examples are computed. These indicate that the error is not serious in the case of fitting a Poisson distribution, but may be so for the fitting of a normal.

Let $x_1 < x_2 < \cdots < x_n$ and $y_1 < y_2 < \cdots < y_m$ be the ordered results of two random samples from populations having continuous cumulative distribution functions $F(x)$ and $G(x)$ respectively. Let $S_n(x) = k/n$, where $k$ is the number of observations of $X$ which are less than or equal to $x$, and $S'_m(x) = j/m$, where $j$ is the number of observations of $Y$ which are less than or equal to $x$. The statistics \begin{align*}d_r &= \underset{x \leqq x_r}\max | S_n(x) - S'_m(x) |, \\ d'_r &= \underset{x \leqq \max(x_r,y_r)}\max | S_n (x) - S'_m(x) |, r \leqq \min (m, n), \\\end{align*} can be used to test the hypothesis $F(x) = G(x)$. For example, using $d_r$ we would reject the hypothesis if the observed $d_r$, that is, the maximum absolute deviation between the two step functions at or below the $r$th observation of a given sample, is significantly large. In this paper, the distributions of $d_r$ and $d'_r$ under the hypothesis $F(x) = G(x)$ are obtained and tabulated. Some possible applications are discussed and a numerical example in life testing is given.

A formula for the power of the analysis of variance test is derived for the case when the denominator of the $F$ ratio has an even number of degrees of freedom. The form employed is particularly adapted to computation of the power as a function of the alternative hypothesis with arbitrary fixed level of significance and fixed degrees of freedom. For $m$ degrees of freedom in the numerator and 2, 4, 6, 8 and 10 in the denominator, the power functions are deduced from the general formula, with an indication of their use.*

Presented are tabulations of the power and power efficiency of four nonparametric tests (rank-sum, maximum deviation, median, and total number of runs) for the difference in means of two samples drawn from normal populations with equal variance. The cases considered are for equal sample sizes of three, four and five observations and alternatives $\delta = | \mu_1 - \mu_2 |/\sigma$.